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On fundamental solutions of binary quadratic form equations

Volume 169 / 2015

Keith R. Matthews, John P. Robertson, Anitha Srinivasan Acta Arithmetica 169 (2015), 291-299 MSC: Primary 11A55, 11D09, 11D45. DOI: 10.4064/aa169-3-4

Abstract

We show that, with suitable modification, the upper bound estimates of Stolt for the fundamental integer solutions of the Diophantine equation $Au^2+Buv+Cv^2=N$, where $A>0$, $N\not =0$ and $B^2-4AC$ is positive and nonsquare, in fact characterize the fundamental solutions. As a corollary, we get a corresponding result for the equation $u^2-dv^2=N$, where $d$ is positive and nonsquare, in which case the upper bound estimates were obtained by Nagell and Chebyshev.

Authors

  • Keith R. MatthewsDepartment of Mathematics
    University of Queensland
    Brisbane 4072, Australia
    and
    Centre for Mathematics and its Applications
    Australian National University
    Canberra, ACT 0200, Australia
    e-mail
  • John P. RobertsonActuarial and Economic Services Division
    National Council on Compensation Insurance
    Boca Raton, FL 33487, U.S.A.
    e-mail
  • Anitha SrinivasanDepartment of Mathematics
    Saint Louis University – Madrid campus
    Avenida del Valle 34
    28003 Madrid, Spain
    e-mail

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